In problem 4, you will be given a
triangle. Use your compass and straightedge to draw in the
perpendicular bisectors of
the three sides, the
bisectors of the three
angles, the three medians, and the three altitudes. To draw a
median, connect a vertex of the triangle with the midpoint of the
opposite side, You will already have the midpoint of the three sides
as a result of having already constructed the
perpendicular bisectors of
the three sides. To draw an altitude, from a vertex of the triangle,
drop a perpendicular to the
opposite side. You should find that the three
perpendicular bisectors meet
at a single point which is the center of the circumscribed circle.
Draw the circle centered at this point that goes through all three
vertices of the triangle. You should also find that the three
angle bisectors all meet
at the same point, This point is the center of the inscribed
triangle. To draw the inscribed triangle, now that you have its
center, you need a point on the circle. To find one,
drop a perpendicular from the
point where the three angle
bisectors meet to one of the sides. The point where the
perpendicular meets the side will be the point of tangency for that
side which will be a point on the circle. Similarly, the three
medians will meet at a single point, which is called the centroid,
and the three altitudes will meet at a single point which is called
the orthocenter. The Euler line goes through the orthocenter, the
centroid, and the center of the circumscribed circle.